Dispersed plug flow model solutions

Comparison of solutions of the axially dispersed plug flow model for different boundary conditions... 

Even with constant dispersion coefficients, accounting for the velocity profile still creates difficulties in the solution of the partial differential equation. Therefore it is common to take the velocity to be constant at its mean value u. With all the coefficients constant, analytical solution of the partial differential equation is readily obtainable for various situations. This model with flat velocity profile and constant values for the dispersion coefficients is called the dispersed plug-flow model, and is characterized mathematically by Eq. (1-4). The parameters of this model are Dr, Dl and u. 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

The pattern of flow through a packed adsorbent bed can generally be described by the axial dispersed plug flow model. To predict the dynamic response of the column therefore requires the simultaneous solution, subject to the appropriate initial and boundary conditions, of the differential mass balance equations for an element of the column,... 

The "axial dispersion" or "axial dispersed plug flow" model [Levenspiel and Bischoff, 1963] takes the form of a one-dimensional convection-diffusion equation, allowing to utilize all of the clasical mathematical solutions that are available [e.g. Carslaw and Jaeger, 1959, 1986 Crank, 1956]. 

Figure El2.2a shows the boundary conditions X0 and Yx. Given values for m, Nox, and the length of the column, a solution for Y0 in terms of vx and vY can be obtained Xx is related to Y0 and F via a material balance Xx = 1 - (Yq/F). Hartland and Meck-lenburgh (1975) list the solutions for the plug flow model (and also the axial dispersion model) for a linear equilibrium relationship, in terms of F ...

First Moments. For both of the dispersed plug-flow cases Mi = 0. This means that the center of gravity of the solute moves with the mean speed of the flowing fluid. For the uniform and the general dispersion models, however, this is not always true. If the solute concentration is initially uniform over a cross-sectional plane, it can be shown (A6) that... 

The plug flow with dispersion model results in a degradation to 3.4% of the inflow trichloroethylene concentration. This is significantly different than the plug flow model (1.0%). It is also a more accurate solution. Whether it is the tail of a tracer pulse or a reaction that approaches complete degradation, one needs to be careful about applying the plug flow model when low concentrations, relative to the inflow, are important. 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

For the dynamic simulation of the SMB-SFC process a plug-flow model with axial dispersion and linear mass-transfer resistance was used. The solution of the resulting mass-balance equations was performed with a finite difference method first developed by Rouchon et al. [69] and adapted to the conditions of the SMB process by Kniep et al. [70]. The pressure drop in the columns is calculated with the Darcy equation. The equation of state from Span and Wagner [60] is used to calculate the mobile phase density. The density of the mobile phase is considered variable. 

It has been shown that the solutions of the plug-flow model lead directly to near-accurate solutions of the axial dispersion model and that the plug-flow model is equivalent to the radial dispersion model with some simplifying assumptions. Radial gradients usually exist when there is a significant heat effect (Chapter 9). For highly exothermic reactions, the usual design practice is to select a small... 

It has been shown in Section 10.3 that the solution of the plug-flow model leads directly to that of the one-dimensional model with axial dispersion. These results can be applied to reactions affected by diffusion. Use of the results leads to the design equations for the fixed-beds with axial dispersion given in Table 10.3. Note that Co is the solution for the plug-flow model and that Eq. (E) in Table 10.3 is used for kb in Eq. (B), whereas Eq. (G) is in Eq. (F) (see Problem 10.9 for the case of = 0). 

Sastri et al. (1983) modeled a three-phase noncatalytic but reactive system to produce industrial concentrations of zinc hydrosulfite (ZnS204) in an SBR. Three different approaches were proposed plug-flow, axial diffusion, and perfect mixing mathematical models. The authors compared the numerical solutions for the three models and noticed that the experimental data are well predicted by the axially dispersed plug-flow (diffusion) model, moderately predicted with the plug-flow model, and poorly predicted with the perfect mixing model. 

Analytical solutions also are possible when T is constant and m = 0, V2, or 2. More complex chemical rate equations will require numerical solutions. Such rate equations are apphed to the sizing of plug flow, CSTR, and dispersion reactor models by Ramachandran and Chaud-hari (Three-Pha.se Chemical Reactors, Gordon and Breach, 1983). 

Levenspiel and Bisehoff [24] eompared the fraetion unreaeted by the dispersion model to the solution with that for plug flow ... 

Solutions with other chemical rate equations are in P8.03.03, and some numerical cases in P8.03.04-P8.03.06. Such rate equations can be applied to the sizing of plug flow, CSTR and dispersion reactor models. 

The assumption that Cou = 1 in equation (6.43) is really only accurate when Pe > 10. The only way to apply this tracer curve to the plug flow with dispersion model while Cou 1 would be to route each portion of the tracer curve through the reactor. With Pe = 9.4, this solution will be close, although stiU an approximation. 

Now, we need a solution to the plug flow with dispersion model for steady-state operation of an air-stripping tower. The mass transport equation for this situation, assuming minimal trichloroethylene builds up in the bubble, is... 

From the appearance of the dispersion number DjuL in this dimensionless form of the basic differential equation of the plug-flow dispersion model it can be inferred that the dispersion number must be a significant characteristic parameter in any solution to the equation, as we have seen. 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

In an ideal fixed-bed reactor, plug flow of gas is assumed. This is, however, not a good assumption for reactive solids, because the bed properties vary with position, mainly due to changing pellet properties (and dimensions in most cases), and hence the use of nonideal models is often necessary. The dispersion model, with all its limitations, is still the most practical one. The equations involved are cumbersome, but their asymptotic solutions are simple, particularly for systems... 

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